Reliability and Survival Methods • Reliability Growth • Additional Examples of the Reliability Growth Platform
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Additional Examples of the Reliability Growth Platform
This section contains two additional examples of the use of the Reliability Growth platform. The first illustrates interval-censored data while the second illustrates the Dates Format for uncensored data.
Piecewise NHPP Weibull Model Fitting with Interval-Censored Data
The sample data file TurbineEngineDesign2.jmp, found in the Reliability subfolder, contains data on failures for a turbine engine design over three phases of a testing program. The first two columns give time intervals during which failures occurred. These intervals are recorded as days since the start of testing. The exact failure times are not known; it is only known that failures occurred within these intervals.
The reports of failures are provided generally at weekly intervals. Intervals during which there were no failures and which fell strictly within a phase are not included in the data table (for example, the interval 106 to 112 is not represented in the table). Because these make no contribution to the likelihood function, they are not needed for estimation of model parameters.
However, to fit a Piecewise Weibull NHPP or Reinitialized Weibull NHPP model, it is important that the start times for all phases be provided in the Time to Event or Timestamp columns.
Here, the three phases began at days 0 (Initial phase), 91 (Revised phase), and 200 (Final phase). There were failures during the weeks that began the Initial and Revised phases. However, no failures were reported between days 196 and 231. For this reason, an interval with beginning and ending days equal to 200 was included in the table (row 23), reflecting 0 failures. This is necessary so that JMP knows the start time of the Final phase.
The test was terminated at 385 days. This is an example of interval-censored failure times with time terminated phases.
Note: The phase start times are required for proper display of the transition times for the Piecewise Weibull NHPP model; they are required for estimation of the Reinitialized Weibull NHPP model. For interval-censored data, the algorithm defines the beginning time for a phase as the start date recorded in the row containing the first occurrence of that phase designation. In our example, if row 23 were not in the table, the beginning time of the Final phase would be taken as 231.
1.
Select Help > Sample Data Library and open Reliability/TurbineEngineDesign2.jmp.
2.
Select Analyze > Reliability and Survival > Reliability Growth.
3.
On the Time to Event Format tab, select the columns Interval Start and Interval End, and click Time to Event.
4.
Select Fixes and click Event Count.
5.
Select Design Phase and click Phase.
6.
Click OK.
7.
From the red triangle menu at Reliability Growth, select Fit Model > Piecewise Weibull NHPP.
The Cumulative Events plot from the Observed Data report is shown in Cumulative Events Plot. The vertical dashed blue lines indicate the phase transition points. The first occurrence of Revised in the column Design Phase is in row 14. So, the start of the Revised phase is taken to be the Interval Start value in row 14, namely, day 91. Similarly, the first occurrence of Final in the column Design Phase is in row 23. So, the start of the Final phase is taken to be the Interval Start value in row 23, namely, day 200.
Cumulative Events Plot
The Piecewise Weibull NHPP report is found under the Models outline node (MTBF Plot). Here we see the mean time between failures increasing over the three phases. From the Estimates report, we see that the estimates of beta decrease over the three testing phases.
MTBF Plot
Piecewise Weibull NHP Change Point Detection with Time in Dates Format
The file BrakeReliability.jmp, found in the Reliability subfolder, contains data on fixes to a braking system. The Date column gives the dates when Fixes, given in the second column, were implemented. For this data, the failure times are known. Note that the Date column must be in ascending order.
The test start time is the first entry in the Date column, 09/29/2011, and the corresponding value for Fixes is set at 0. This is needed in order to convey the start time for testing. Had there been a nonzero value for Fixes in this first row, the corresponding date would have been treated as the test start time, but the value of Fixes would have been treated as 0 in the analysis.
The test termination time is given in the last row as 05/31/2012. Because the value in Fixes in the last row is 0, the test is considered to be time terminated on 5/31/2012. Had there been a nonzero value for Fixes in this last row, the test would have been considered failure terminated.
1.
Select Help > Sample Data Library and open Reliability/BrakeReliability.jmp.
2.
Select Analyze > Reliability and Survival > Reliability Growth.
3.
Select the Dates Format tab.
4.
Select Date and click Timestamp.
5.
Select Fixes and click Event Count.
6.
Click OK.
7.
From the red triangle menu at Reliability Growth, select Fit Model > Crow AMSAA.
The Cumulative Events plot in the Observed Data report updates to show the model (Cumulative Events Plot with Crow AMSAA Model). The model does not seem to fit the data very well.
Cumulative Events Plot with Crow AMSAA Model
8.
From the red triangle menu at Reliability Growth, select Fit Model > Piecewise Weibull NHPP Change Point Detection.
The Cumulative Events plot in the Observed Data report updates to show the piecewise model fit using change-point detection. Both models are shown in Cumulative Events Plot with Two Models. Though the data are rather sparse, the piecewise model appears to provide a better fit to the data.
Cumulative Events Plot with Two Models